CHAPTER LXII — Critical Commentary & Notes

On the Way to Measure All Kinds of Columns, and First the Round Columns (Cylinders).

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagrams: Folio 0054 (f. 19v) includes the unit cube diagram 0054.g7.png and the circle inscribed in the square \(abcd\) with diameter \(ef\) and center \(g\) (0054.g8.png), visually showing why the circle occupies \(\frac{11}{14}\) of the circumscribed square and the four corner spandrels occupy \(\frac{3}{14}\).

  • Archimedes’ *Measurement of a Circle*: Proposition 3 of Archimedes’ :cite:`Dimensio Circuli <archimedes:dimensio-circuli>` proves \(3\frac{10}{71} < \pi < 3\frac{1}{7}\), whence Proposition 2 states that the area of a circle to the square on its diameter is very nearly \(11 : 14\).

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnotes 1–4 (Cap. LXII, pp. 256–257): (1) Namely, solid bodies (scil. Körper). (2) Compare the marginal figures on pp. 104 and 108 (Paganini Folios 0051–0053). (3) Compare the first figure on p. 110 (Paganini Folio 0053, 0053.g2.png). (4) Supply the missing verb ‘proved’ (bewiesen) in Pacioli's sentence on Archimedes’s quadrature of the circle. (Winterberg Fussnoten 1–4 (Cap. LXII, S. 256–257): 1) scil. Körper. 2) vgl. die Figuren pag. 104 und 108. 3) vgl. 1. Fig. pag. 110. 4) Das im Text fehlende Verbum „bewiesen“ zu ergänzen.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0053, 0054 (f. 19r–f. 19v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/062_cap_LXII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/062_cap_LXII.txt

  • Marginal Woodcut Diagrams: 5 figure(s) extracted from the 1509 Venice witness.